The home side forged into a 24-3 half-time lead and staved off a mini-revival from Selby midway through the second half to claim a comfortable win.
They broke through in the second minute, Russ Downey touching down for John Dawes to convert.
Selby’s Josh Cruise missed a penalty before Dawes made it 10-0 with a 14th-minute kick.
Dawes added further penalties on 21 minutes and 25 minutes before York made it 21-0 with a 28th-minute try from Toby Atkin after a Downey break.
Cruise got the Swans on the board with a 32nd minute penalty, but Dawes made it 24-3 with a kick in first-half injury time.
Selby’s Duncan Hardy was sin-binned early in the second half and Dawes extended the lead with a penalty.
The York kicker then touched down in the 55th minute for a try he also converted.
With Dawes in the sin-bin, Selby’s Kevin Lamb crossed in the 59th minute and Cruise then convertted a 64th-minute score from Martin Protheroe.
Dawes returned on 70 minutes, but failed to add the extras to a Charlie Nicholson try and then missed a 76th-minute penalty.
The Ryedale side trailed 22-7 early in the second half but fought back to lead 24-22 in the closing stages, only to concede a try.
Flanker Ali Coe was Malt’s first-half try scorer, with Charlie Ramsay converting.
After the break they responded with two tries by centre Kenton Leiataua, both converted by Ramsay, who also kicked a penalty.
Pocklington’s poor run of results continued as they lost 33-13 at Alnwick.
Though Pocklington led 8-5 ten minutes into the second half, they were then lacerated by a four-try salvo from the free running home threequarters. However, the damage had been done in the first period when Pocklington failed to capitalise on the strong wind behind them.
Pock went ahead early on through a Jack Fletcher try, but they spent long periods in Alnwick territory without really threatening the home line.
Alnwick’s backs always looked dangerous on the counter and they grabbed a shock score with the last play of the half when they followed up to touch down after a kick ahead had bobbled loose.
Pocklington made a lively start to the second period and when flanker Jack Holbrough charged into the Alnwick 22, number eight Marcus Britland took over go through the maul and his show and go opened the way to canter 20 yards to score.
Pock’s lead was short lived as Alnwick scored twice in three minutes to turn the tables.
Alnwick rubbed it in with two more back tries in the last quarter before Pocklington finished with a late score, running the ball from deep in their own half for replacement centre Tim Nixon to break clear and hand on to wing Charlie Allen, who finished well.
York RI lost 38-13 at Leeds Corinthians in Yorkshire Four – but the final scoreline flattered the hosts.
Leeds scored three tries in the first half to lead 21-3, with Stuart Watson kicking a penalty for the visitors.
Against the run of play, the home side kicked a penalty and got another converted try to lead 31-3.
RI upped their game and the forwards drove over for Rob Agen to score, then a great break saw Chris Morley pull it back to 31-13.
With RI pressing at the death, the Leeds outfit broke away to end the game with a further try.
Rob Cuthbertson put in a flawless display for RI.